The equivalence of pseudodifferential operators and their symbols via \v{C}ech-Dolbeault cohomology

TL;DR

This paper establishes an isomorphism between pseudodifferential operators and their symbols using cech-Dolbeault cohomology and introduces a new symbol class called symbols of $C^infty$-type.

Contribution

The paper introduces a novel approach to construct the sheaf morphism between pseudodifferential operators and symbols via cech-Dolbeault cohomology and defines a new symbol class.

Findings

Constructed a sheaf morphism from pseudodifferential operators to their symbol class.

Proved the sheaf morphism is an isomorphism.

Introduced the symbols of $C^infty$-type as a new symbol class.

Abstract

In this paper we construct the sheaf morphism from the sheaf of pseudodifferential operators to its symbol class. Since the map is hard to construct directly, we realize it with two original ideas as follows. First, to calculate cohomologies we use the theory of \v{C}ech-Dolbeault cohomology introduced by Honda, Izawa and Suwa. Secondly we construct a new symbol class, which is called the symbols of $C^\infty$-type. These ideas enable us to construct the sheaf morphism, which is actually an isomorphism of sheaves.